Some new results on mutually orthogonal frequency squares

Let n = λ m . A frequency square F ( n ; λ ) is an n × n array in which each of m distinct symbols appears exactly λ times in each row and column. Two such squares are said to be orthogonal if upon superposition, each of the m 2 distinct ordered pairs occurs exactly λ 2 times. Hedayat, Sloane and Stufken in their book Hedayat et al. (1999) provided a small table of lower bounds for the maximum number of mutually orthogonal frequency squares of type F ( n ; λ ) (HSS Table), and posed the following research problem: Improve the lower bound for the entries in HSS Table. e and Mullen in 2001 extended the table (LM Table). In this article we will give some new construction methods of mutually orthogonal frequency squares. As an application we improve the lower bounds for more than half of the entries in LM Table.